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a(n) is the number of interior points over all Motzkin polyominoes of length n.
1

%I #6 Jan 22 2024 00:03:27

%S 1,6,25,93,324,1088,3565,11487,36564,115327,361194,1124932,3487857,

%T 10774350,33181365,101924727,312402600,955728009,2919095808,

%U 8903220174,27121002621,82525328253,250867833102,761946207338,2312406692839,7012910929818,21254741084817,64381661704933

%N a(n) is the number of interior points over all Motzkin polyominoes of length n.

%H Jean-Luc Baril, Sergey Kirgizov, José L. Ramírez, and Diego Villamizar, <a href="https://arxiv.org/abs/2401.06228">The Combinatorics of Motzkin Polyominoes</a>, arXiv:2401.06228 [math.CO], 2024. See Corollary 6.4 at pages 18-19.

%F a(n) = (3^n - 3*A002426(n))/2 - 2*A002426(n-1) + 2*A001006(n-1).

%t t[n_]:=SeriesCoefficient[1/(Sqrt[1-2x-3x^2]),{x,0,n}]; (* A002426 *)

%t m[n_]:=SeriesCoefficient[(1 - x - (1 - 2x - 3x^2)^(1/2))/(2x^2), {x, 0, n}]; (* A001006 *)

%t a[n_]:=(3^n-3t[n])/2-2t[n-1]+2m[n-1]; Array[a,28,3]

%Y Cf. A001006, A002426, A369359.

%K nonn

%O 3,2

%A _Stefano Spezia_, Jan 21 2024